ECONOMIES OF SCALE FOR DATA ENVELOPMENT ANALYSIS WITH A KANSAS FARM APPLICATION BRYON JAMES PARMAN

Transcription

1 ECONOMIES OF SCALE FOR DATA ENVELOPMENT ANALYSIS WITH A KANSAS FARM APPLICATION by BRYON JAMES PARMAN B.A., Peru State College, 2008 M.S., University of Nebraska-Omaha, 2010 AN ABSTRACT OF A DISSERTATION Submitted in partial fulfillment of the requirements for the degree DOCTOR OF PHILOSOPHY Department of Agricultural Economics College of Agriculture KANSAS STATE UNIVERSITY Manhattan, Kansas 2013

2 Abstract Estimation of cost functions can provide useful economic information to producers, economists, and policy makers. From the estimation of a cost function, it is possible to calculate cost efficiency, economies of scope, and economies of scale. Economic theory specifies the cost function as a frontier since firms cannot operate at lower cost than the cost minimizing input/output bundle. However, traditional parametric estimation techniques often violate economic theory using two sided-error systems. The stochastic frontier method has allowed the estimation of a frontier but continues to restrict the technology through functional assumption. Nonparametric frontier estimation is an alternative approach to estimate a cost frontier by enveloping the data which by its construct, conforms to economic theory. This research expands the economic information available by deriving multi-product scale economies and productspecific scale economies from the nonparametric approach. It also tests its ability to accurately recover theses important economic measures under different assumptions of the cost function, and cost inefficiency distributions. Next, this new method is compared to other methods used to estimate cost functions and associated economic measures including a two-sided error system, stochastic frontier method, and an OLS model restricting the errors to take on only positive values. Finally, the nonparametric approach with the new measures is applied to a sample of Kansas farms. The nonparametric approach is able to closely estimate economies of scale and scope from estimation of a cost frontier. Comparison reveals that the nonparametric approach is closer to the true economic measures than some parametric methods and that it is better able to extrapolate out of sample when there are no zero output firms. Finally, the nonparametric

3 approach shows that potential cost savings from economies of scale and economies of scope exist for small Kansas farms. However, cost savings from economies of scale become exhausted when farms exceed gross annual revenues of $500k, while economies of scope also diminish as farms grow larger. Results also show from annual frontier estimations that estimates of economies of scale, scope, and cost efficiency have remained relatively stable from 2002 to 2011.

4 ECONOMIES OF SCALE FOR DATA ENVELOPMENT ANALYSIS WITH A KANSAS FARM APPLICATION by BRYON JAMES PARMAN B.A., Peru State College, 2008 M.S., University of Nebraska-Omaha, 2010 A DISSERTATION Submitted in partial fulfillment of the requirements for the degree DOCTOR OF PHILOSOPHY Department of Agricultural Economics College of Agriculture KANSAS STATE UNIVERSITY Manhattan, Kansas 2013 Approved by: Co-Major Professor Vincent Amanor-Boadu Approved by: Co-Major Professor Allen M. Featherstone

5 Copyright BRYON JAMES PARMAN 2013

6 Abstract Estimation of cost functions can provide useful economic information to producers, economists, and policy makers. From the estimation of a cost function, it is possible to calculate cost efficiency, economies of scope, and economies of scale. Economic theory specifies the cost function as a frontier since firms cannot operate at lower cost than the cost minimizing input/output bundle. However, traditional parametric estimation techniques often violate economic theory using two sided-error systems. The stochastic frontier method has allowed the estimation of a frontier but continues to restrict the technology through functional assumption. Nonparametric frontier estimation is an alternative approach to estimate a cost frontier by enveloping the data which by its construct, conforms to economic theory. This research expands the economic information available by deriving multi-product scale economies and productspecific scale economies from the nonparametric approach. It also tests its ability to accurately recover theses important economic measures under different assumptions of the cost function, and cost inefficiency distributions. Next, this new method is compared to other methods used to estimate cost functions and associated economic measures including a two-sided error system, stochastic frontier method, and an OLS model restricting the errors to take on only positive values. Finally, the nonparametric approach with the new measures is applied to a sample of Kansas farms. The nonparametric approach is able to closely estimate economies of scale and scope from estimation of a cost frontier. Comparison reveals that the nonparametric approach is closer to the true economic measures than some parametric methods and that it is better able to extrapolate out of sample when there are no zero output firms. Finally, the nonparametric

7 approach shows that potential cost savings from economies of scale and economies of scope exist for small Kansas farms. However, cost savings from economies of scale become exhausted when farms exceed gross annual revenues of $500k, while economies of scope also diminish as farms grow larger. Results also show from annual frontier estimations that estimates of economies of scale, scope, and cost efficiency have remained relatively stable from 2002 to 2011.

8 Table of Contents List of Figures... xi List of Tables... xiii Acknowledgements... xv Chapter 1 - Introduction... 1 Research Motivation... 4 Chapter 1 References... 8 Chapter 1 Figures... 9 Chapter 2 - A Nonparametric Approach to Multi-product and Product-specific Economies of Scale Introduction Theory Data and Methods The Nonparametric Method Data Simulation Results Multi-product Economies of Scale Product-specific Economies of Scale Economies of Scope Cost Efficiency Conclusions Chapter 2 References Chapter 2 Tables Chapter 2 Figures Chapter 3 - A Comparison of Parametric and Nonparametric Estimation Methods for Cost Frontiers and Economic Measures Introduction Data Estimation Methods viii

9 The Two-Sided Error System Equation The OLS Estimator with Positive Errors The Stochastic Frontier Cost Function Estimator The Nonparametric Approach Results Cost Efficiency Economies of Scope Multi-product Economies of Scale Product-Specific Economies of Scale Implications of the Results Conclusions Chapter 3 References Chapter 3 Tables Chapter 3 Figures Chapter 4 - A Nonparametric Approach to Multi-product and Product-specific Scale Economies, Economies of Scope, and Cost Efficiency for Kansas Farms Introduction Methods Data Results Cost Efficiency Multi-product Economies of Scale Economies of Scope Product-specific Economies of Scale Implications Differences between Annual Frontier and Single Frontier Analysis Implications for KFMA Farms Conclusions Chapter 4 References Chapter 4 Tables Chapter 4 Figures ix

10 Chapter 5 - Conclusions Chapter 5 References Appendix A - Software Codes Used in Chapter A.1 Data Generation Using Shazam Software Package A.2 Nonparametric Estimation Using General Algebraic Modeling Software (GAMS) Appendix B - Software Codes Used in Chapter B.1 Two-Sided Error Model B.2 OLS Positive Errors Model B.3 Stochastic Frontier Model B.4 Nonparametric Model Appendix C - Software Codes Used in Chapter C.1 Software Code for Cost Frontier Estimation Using KFMA Data x

11 List of Figures Figure 1.1 Actual Total Cost for Three Firms and the Total Cost Frontier... 9 Figure 1.2 Minimum Average Total Cost and Actual Average Total Cost for Three Firms Figure 2.1 Frontier Multi-Product Scale Economies Cumulative Frequency for Generated Data Figure 2.2 Frontier Product-Specific Scale Economies Figure 2.3 Frontier Economies of Scope Cumulative Frequency Figure 2.4 Frontier Cost Efficiencies Cumulative Frequency for both Half-normal and Uniform Distributions Figure 2.5 Differences between frontier Multiproduct Scale Economies and nonparametric estimates of Multiproduct Economies of Scale for Half-normal and Uniform Cumulative Distributions Figure 2.6 Differences between frontier Product-specific Economies of Scale for Y1 and nonparametric estimates of Product-specific Economies of Scale for Y1 for Half-normal and Uniform Cumulative Distributions Figure 2.7 Differences between frontier Product-specific Economies of Scale for Y2 and nonparametric estimates of Product-specific Economies of Scale for Y2 for the Half-normal and Uniform Cumulative distributions Figure 2.8 Differences between frontier Product-specific Economies of Scale for Y1 and nonparametric estimates of Product-specific Economies of Scale for Y1 removing technical inefficiency from frontier firms Figure 2.9 Differences between frontier Economies of Scope and nonparametric estimates of Economies of Scope for Half-normal and Uniform distributions Figure 2.10 Differences between frontier Cost Efficiency and nonparametric estimates of Cost Efficiency for Half-normal and Uniform Cumulative Distribution Figure 3.1 Frontier Multi-Product Scale Economies Cumulative Frequency for Simulated Data Figure 3.2 Frontier Cost Efficiencies Cumulative Frequency for both Half-normal and Uniform Distributions xi

12 Figure 3.3 Frontier Economies of Scope Cumulative Frequency Figure 3.4 Frontier Product-Specific Scale Economies Figure 3.5 Differences between frontier cost efficiency and estimated cost efficiency for the nonparametric, frontier, and OLS positive errors models Figure 3.6 Differences between frontier Economies of Scope and estimated Economies of Scope from Two-sided Errors, OLS Positive Errors, Frontier, and Nonparametric models Figure 3.7 Differences between frontier Multi-product Scale Economies and estimated Multiproduct Scale Economies from the Two-sided Errors, OLS Positive Errors, Frontier, and Nonparametric models Figure 3.8 Differences between frontier Product-specific Scale Economies for Y 1 and estimated Product-specific Scale Economies for Y 1 from the Two-sided Errors, OLS Positive Errors, Frontier, and Nonparametric models Figure 3.9 Differences between frontier Product-specific Scale Economies for Y 2 and estimated Product-specific Scale Economies for Y 2 from the Two-sided Errors, OLS Positive Errors, Frontier, and Nonparametric models Figure 4.1 Cumulative Density of Cost Efficiency Estimates for Kansas Farms Categorized by Farm Gross Revenue Figure 4.2 Cumulative Density of Multi-product Scale Economies Estimates for Kansas Farms Categorized by Farm Gross Revenue Figure 4.3 Cumulative Density of Economies of Scope Estimates for Kansas Farms Categorized by Farm Gross Revenue Figure 4.4 Cumulative Density of Crop-specific Scale Economy Estimates for Kansas Farms Categorized by Farm Gross Revenue Figure 4.5 Cumulative Density of Livestock-specific Scale Economy Estimates for Kansas Farms Categorized by Farm Gross Revenue xii

13 List of Tables Table 2.1 Coefficients used in cost function for data simulation for half-normal and uniform distributions Table 2.2 The average, standard deviation, minimum and maximum for the input/output quantities and input prices in half-normal (x n i ) and uniform (x u i ) cases Table 2.3 Summary statistics for MPSE, PSE, economies of scope, and cost efficiency Table 2.4 Statistics for simulated multi-product scale economies estimates minus multi-product scale economies estimated nonparametrically for the half-normal and uniform distributions Table 2.5 Statistics for simulated product-specific scale economies estimates minus productspecific scale economies estimated nonparametrically for the half-normal and uniform distributions for outputs 1 and Table 2.6 Statistics for simulated product-specific scale economies estimates minus productspecific scale economies estimated nonparametrically for y 1 removing the technical inefficiency in the input quantities Table 2.7 Statistics for simulated scope economies estimates minus scope economies estimated nonparametrically Table 2.8 Statistics for simulated cost efficiency minus cost efficiencies estimated nonparametrically for half-normal and uniform distributions Table 3.1 Assumed coefficients used in cost function for data simulation for half-normal and uniform distributions Table 3.2 The average, standard deviation, minimum and maximum for the input/output quantities and input prices in half-normal (x n i ) and uniform (x u i ) cases Table 3.3 Summary statistics for efficiency calculations from generated data including halfnormal and uniform distributions Table 3.4 Parameter estimates and standard errors for three simulations of each parametric model Table 3.5 Eigenvalues for B (prices) and C (outputs) matrices for each model and simulation xiii

14 Table 3.6 Statistics for simulated cost efficiency differences for the OLS positive errors, stochastic frontier, and nonparametric estimations Table 3.7 Statistics for economies of scope differences from all four methods from all three data sets Table 3.8 Statistics for Multi-product Scale Economies differences from all four methods from all three data sets Table 3.9 Statistics for Product-specific Scale Economies differences for y 1 and y 2 from all four methods and all three data sets Table 4.1 Summary statistics for Kansas Farm Management Farms of input and output quantity indices, 2002 to Table 4.2 Price indices for farm inputs and outputs for each year Table 4.3 Overall summary statistics for estimated cost measures for Kansas Farm Management Farms estimated from a single frontier and annually Table 4.4 Annual averages for cost efficiency, MPSE, PSEs, and economies of scope for Kansas Farm Management Farms Table 4.5 F-Test results evaluating statistical differences in cost frontiers Table 4.6 Summary statistics for cost efficiency for Kansas Farm Management Farms estimated from a single frontier and annually Table 4.7 Summary statistics for multi-product economies of scale for Kansas Farm Management Farms estimated from a single frontier and annually Table 4.8 Summary statistics for economies of scope from Kansas Farm Management Farms estimated from a single frontier and annually Table 4.9. Summary statistics for crop-specific economies of scale categorized by gross revenues estimated simultaneously and individually by year Table 4.10 Summary statistics for livestock-specific economies of scale categorized by gross revenues estimated simultaneously and individually by year xiv

15 Acknowledgements This material is based upon work supported by National Science Foundation Grant # : Integrating the Socioeconomic, Technical, and Agricultural Aspects of Renewable and Sustainable Biorefining Program, awarded to Kansas State University. Funding by the Center for Sustainable Energy, Kansas State University is kindly acknowledged I would first like to thank Dr. Vincent Amanor-Boadu for his sound advice, guidance, and mentoring as I worked toward completion of my degree. For me, the latitude and encouragement he has provided allowed me to pursue my own research interests with the opportunity to work on multiple projects over the years. I am very grateful for the conversations we have had which have helped give me perspective in many areas I had not previously considered. I would like to thank Dr. Allen Featherstone for recently setting me on a research path that I have found interesting and challenging, and believing that I was capable of completing this latest research. I would like to acknowledge all of his hard work and patience during the development of this dissertation, and I am especially grateful for the very long hours he has put in editing and refining the work contained herein. To my committee members, Dr. Jason Bergtold, and Dr. Todd Easton I would like to thank them both for helping me through this process and for their advice on my research. I would like to especially thank Dr. Peter Pfromm for his advice and encouragement for not only this research, but on previous work and associated presentations conducted during pursuit of my degree. I am also very grateful to Dr. Mary Rezac and Keith Rutlin for accepting me into the IGERT program at Kansas State University. They were instrumental in providing the resources I xv

16 needed to complete my goals and my degree and provided some unique opportunities for travel and scholarly collaboration not available to many graduate students in my position. xvi

17 Chapter 1 - Introduction The economic definition of a cost frontier is that it represents the lowest cost for producing a given level of output. There are fundamental elements in the study and evaluation of industry structure. Firms on the frontier change their cost by changing their output level or output bundle. These firms are unable to improve cost through alterations to their input mix. Firms above the frontier are not efficient and can reduce their cost by changing the output levels or their input bundle. Figure 1.1 illustrates a single output cost frontier where the cost curve is the minimum cost to produce a given output level. Points A, B, and C represent the actual total cost for three firms where firms A and B are producing at costs higher than the frontier cost for their respective output levels. Point C is operating on the cost frontier. The calculation of cost efficiency (CE i ) of a firm, i, is the ratio of minimum cost (TC min ) to actual total cost incurred in the production of the output (ATC i ) and represents the distance the firm is from the frontier. CE i TC ATC min (1.1) i When estimating the cost frontier, economies of scale for firm i can be calculated. Economies of scale refer to the cost reductions obtained as the firms size approaches constant returns to scale (Figure 1.2). The economies of scale of firm i in the production of output Y, (S iy ) may be determined for a single product Y, produced by the firm as follows: S iy CY ( ) CY ( ) Y Y (1.2) 1

18 where C(Y) is the total cost and C(Y)/ Y is the marginal cost for producing Y. Figure 1.2 represents the average cost of firms A, B, and C. Firm C is on the frontier operating at the minimum average total cost, yielding a scale economy measure equal to one and a cost efficiency measure equal to one. Firms A and B are off the frontier by distance α A and α B, respectively such that both firms can reduce average total cost by moving closer to the frontier. However, firm A can reduce costs more by exploiting economies of scale rather than improving cost efficiency. The measure θ A is the distance from the frontier to the line tangent to the minimum average total cost that defines potential savings from increasing output. Since θ A > α A, holding CE (α A ) constant and increasing output to the same level as firm C reduces average cost more than improving cost efficiency. Firm B, since θ B < α B, improves cost efficiency while holding output constant leading to a reduction in its average total cost more than through output growth. For the case of a multi-output firm, economies of scale (MPSE) for a firm producing i products, is defined as follows: MPSE i CY ( ) C(Y) Yi Y i (1.3) The cost frontier for multi-output firms allows the calculation of product-specific economies of scale (PSE) and economies of scope as well. PSEs are calculated holding all other outputs constant while examining cost as one of the other outputs is varied. The calculation of PSE uses the marginal cost in addition to the incremental cost and average incremental cost for 2

19 the output of interest. The incremental cost (IC i ) represents the cost the firm would incur were it to produce only output i. That is: ICi( Y ) C(Y) C(Y Ni) (1.4) where Y N-i is a vector with a zero component in place of Y i and components equal to Y elsewhere. The average incremental cost (AIC i ) is the incremental cost divided by the output AIC i IC Y i (1.5) i Product-specific scale economies for firm i are defined as the ratio of AIC and marginal cost: PSE i AIC CY ( i ) Y i (1.6) Economies of scope (SC) are the potential cost savings that exist from simultaneous production of more than a single output by a single firm. Economies of scope measure the relative increase in cost should the firm split and produce each product individually. Mathematically, economies of scope for product Y is: CY ( T) CY ( NT) CY ( ) SC( Y) (1.7) CY ( ) where C(Y T ) and C(Y N-T ) respectively define the cost of producing product Y T and the remaining products Y N-T. Multi-product economies of scale are a function of product-specific economies of scale and economies of scope. The relationship between multi-product scale economies (MPSE), 3

20 product-specific scale economies (PSE), and economies of scope (SC) can be determined by defining: i CY ( ) Yi Y i 2 CY ( ) Yi Y i i1 (1.8) Where α i is the weight placed on the PSE of interest based upon its relative contribution to total output from a two output firm: ipsei( Y ) (1 i) PSENi( Y ) MPSE 1 SC( Y ) (1.9) MPSE can take one of three values: decreasing, constant or increasing returns to scale. Equation 1.9 allows factors underlying the measures of MPSE. If SC(Y) is zero and the numerator is less than 1, equal to 1 or greater than 1, then there are decreasing, constant and increasing returns to scale. If SC(Y) is greater than zero and the PSEs are at constant returns to scale, MPSE is increasing (>1). Research Motivation Historically, cost frontiers were econometrically estimated assuming a functional form such as a translog (Christensen et. al. 1973), normalized quadratic (Diewert and Wales 1988), or Generalized Leontief (Diewert 1971) using standard two-sided error systems where some errors are positive (above the frontier), and others are negative (below the frontier). Multiproduct-scale economies, product-specific scale economies, and economies of scope can be estimated from the parameter estimates. This approach allows firms to operate at lower costs than the cost frontier. 4

21 Frontier estimation research has addressed the issue of two-sided error estimations by ensuring that frontiers do not have firms below the frontier using both parametric and nonparametric approaches. The parametric approach is the stochastic frontier method by Aigner, Lovell and Schmidt with a nonparametric alternative, the data envelopment analysis method (DEA), proposed by Farrell. Frontier approaches have typically analyzed the relative efficiency of firms in relationship to the frontier. Measures of economies of scale and/or scope are not typically reported. Thus, economic analysis has focused on firms above the cost frontier or the behavior of the frontier using dual methods and not both as illustrated in Figures 1.1 and 1.2. This dissertation will examine methods to unify the measurement of scope, scale, and cost efficiency. The stochastic error methods can be used to measure scope and scale, but have typically focused on cost efficiency. The nonparametric method (DEA) has also focused primarily on cost efficiency, although Chavas and Aliber propose a method for measuring economies of scope. Measures of product-specific economies of scale and multi-product economies of scale have not been formalized in the literature for the nonparametric method. Finally, the literature does not contain analysis that compares the accuracy of alternative techniques to measures of cost efficiency, economies of scale and economies of scope from a true cost frontier. This dissertation has been organized into three papers. The first formalizes and tests a method for calculating multi-product and product-specific scale economies from an estimated nonparametric cost frontier. This contributes to the literature by increasing the amount of information that may be reported from nonparametric estimation. The analysis uses two datasets 5

22 assuming a true cost function generated using Monte Carlo. The simulations are conducted using half-normal and uniform distributions of cost efficiency. This allows for a comparison between the two distributions of the errors for the true cost frontier. Two true cost functions were assumed as well and two data sets examined with one containing single output firms and multiple output firms, and one with only multiple output firms. The second study compares the accuracy of the nonparametric approach developed in the first paper to three parametric approaches. The three parametric approaches evaluated are a twosided error system, the stochastic frontier method, and an OLS model in which all errors are restricted to be positive. The data sets used for the first objective are used for the second comparison. The different models abilities to accurately calculate the cost efficiency, economies of scope, multi-product economies of scale, and product-specific economies of scale measures are then evaluated. The final paper uses the methods and techniques developed and tested under the first two papers on farm level data instead of simulated data. The data are obtained from the Kansas Farm Management Association for 241 farms from 2002 to Under this objective, the foregoing methods are used to estimate the cost efficiency measures along with multi-product and productspecific scale economies, and economies of scope for these Kansas farms. The dissertation is presented in the next three chapters. The information presented in these chapters will be useful to researchers, economists, managers, and policymakers as it provides sound economic tools for quantifying the cost advantages farms have due to their relative size. The tools allow the determination of the extent that small farms may improve their cost savings from increasing output exploiting economies of scale. It will then be determined 6

23 how much costs are reduced by producing multiple outputs (scope) rather than each output individually. In addition, cost efficiency reveals the reduction in overall costs that can be obtained by appropriately adjusting the input mix. 7

24 Chapter 1 References Aigner, D.J., C.A.K. Lovell, and P. Schmidt. Formulation and estimation of Stochastic Frontier Production Models. Journal of Econometrics, 6(1977): Baumol, William J., John C. Panzar, and Robert O. Willig. Contestable Markets and the Theory of Industry Structure, Harcourt Brace Javanovich, Inc., New York, NY Chavas, Jean-Paul and M. Aliber. An Analysis of Economic Efficiency in Agriculture: A Nonparametric Approach. Journal of Agricultural and Resource Economics, 18(1993): Christensen, Lauritis R. Dale W. Jorgenson, and Lawrence J. Lau. Transcendental Logarithmic Production Frontiers The Review of Economics and Statistics, 55,1(1973): Diewert, W.E. An Application of the Shephard Duality Theorem: A Generalized Leontief Production Function Journal of Political Economy, 79,3(1971): Diewert, W.E. and T.J. Wales. A Normalized Quadratic Semiflexible Functional Form Journal of Econometrics, 37, 3(1988): Färe, R., S. Groskopf, and C.A.K Lovell. The Measurement of Efficiency of Production, Boston: Kluwer-Nijhoff, Farrell, M. J. The Measurement of Productive Efficiency. Journal of the Royal Statistical Society. 120, 3(1957):

25 Chapter 1 Figures Minimum Total Cost Frontier Figure 1.1 Actual Total Cost for Three Firms and the Total Cost Frontier 9

26 Figure 1.2 Minimum Average Total Cost and Actual Average Total Cost for Three Firms 10

27 Chapter 2 - A Nonparametric Approach to Multi-product and Product-specific Economies of Scale Introduction Producer theory provides useful tools for exploring the structure of cost. Estimates of frontier functions, and the distance of firms from the frontier provide insights into how firms with similar technological access and marketing achieve different levels of production efficiency and average costs. These methods allow firms operating off the frontier to understand the potential disadvantages due to sub-optimal output and input bundling choices and the effects on firm performance. Traditionally, multi-product and product-specific economies of scale and economies of scope are estimated parametrically using two-sided error systems through specification of a cost function and estimation of parameters (Christenson et. al.). The error structure is important in the estimation of a cost frontier function since negative errors imply that some firms are actually producing at a lower cost or higher quantities than the frontier that was being estimated which is not consistent with the economic definition of a cost function (Farrell 1957). The stochastic frontier method has addressed the concerns of two-sided error systems by restricting the errors using positive error models (Aigner, Lovell, and Schmidt). Lusk et al. examined the relative variability needed in the estimation of dual cost functions. They found that the relative variability necessary to accurately estimate a dual cost function requires more than 20 years of data based on observations. Thus, dual cost functions may have difficulty recovering the underlying technology. Featherstone and Moss note that parametric frontier estimations may also violate curvature of the cost function. Therefore, the 11

28 lack of ability to accurately measure the underlying technology given data availability and the frontiers not maintaining the necessary cost function conditions are issues that may affect parametric frontier methods. One of the methods used for frontier estimation is the nonparametric method that constructs a frontier from a series of line segments using a linear cost minimization program (Färe, Groskopf, and Lovell). With this method, it is not necessary to restrict the production technology by imposing a functional form. The nonparametric approach conforms to economic theory because curvature restrictions on the production/cost function are imposed in the estimation process. Further, the nonparametric method of Färe et al. may allow technology to be measured using a single year s data; thus, reducing the need of relative price variability to accurately measure technology using the dual approach. Numerous studies have used nonparametric methods to analyze efficiency in various industries including Banker and Maindiratta, Jaforullah and Whiteman, and Chavas and Cox. In these studies, several types of efficiencies are estimated to determine if a firm is producing on the production or cost frontier, whether the firm is optimally allocating inputs, or if the firm is operating at the most efficient size. Chavas and Aliber measure scope economies to determine cost savings from production portfolio diversification in the nonparametric framework. Typical parametric measures of multi-product scale and product-specific scale measures have not yet been developed in the nonparametric DEA framework. For example, Paul et. al. (2004) and Kumar and Gulati (2008) use the DEA method to estimate scale efficiency which takes on values of less than, equal to, or greater than one giving an indication of returns to scale. This measure follows from Ray (1998) and Cooper et. al. (2007) where the DEA method is estimated assuming constant returns to scale, and then again assuming variable returns to scale 12

29 and takes the ratio of the two measures. However, Paul et al. explain that the interpretation of scale efficiency is not as straight forward as a traditional scale economy measure explained by Baumol et. al. Specifically, they note that these measures only indicate if average per-unit costs are increasing, decreasing, or constant, but not necessarily the magnitude of cost savings from scaling. In both Paul et. al., and Kumar and Gulati, it was necessary to perform a parametric estimation to recover traditional estimations of economies of scale, and compare the results to their DEA estimation. Further, techniques for estimating product-specific economies of scale have not been reported for the nonparametric method. This research develops and tests estimation techniques for multi-product and productspecific economies of scale for the nonparametric method. Specifically, this research develops a multi-product and product-specific scale measure using the definition of Baumol et. al. from nonparametrically estimated marginal costs, incremental costs, and output quantities. The estimated measures are then compared to an assumed known cost frontier. From this comparison, it is possible to assess the accuracy of the nonparametric approach estimates and proposed economic measures. In addition, previous research that estimates economies of scope with the nonparametric approach has dropped one or more of the output constraints when estimating the cost of producing a single output (Chavas and Aliber). This research examines that procedure by comparing a method which requires that output to be zero as required in the theory of an incremental cost. The principle advantage to forcing the output to zero rather than dropping it is that it should more closely measure the theoretically defined incremental cost of each output. Further, we evaluate the nonparametric approach under alternative efficiency distributions to investigate the robustness of the results. 13

30 Theory Typical economic measures calculated from a cost function estimation include economies of scale and economies of scope. Measures of scale economies include both multi-product economies of scale (MPSE), and product-specific scale economies (PSE) differing only in that MPSE refers to changes in cost relative to more than one output in a multi-output firm, while PSE refers to proportionate changes in cost relative to a single output (Baumol et al.). Mathematically these measures are defined as follows where C(Y) represents the cost of production with C(Y)/ Y p representing the marginal cost of the p th output. MPSE p CY ( ) C(Y) Yp Yp (2.1) To calculate PSE, the average incremental cost (AIC p ) of producing p must be calculated where the incremental cost (IC) for the p th output is defined as: IC C C j (2.2) p p j p j Thus, AIC p IC p (2.3) y p Product-specific economies of scale are the ratio of the average incremental cost of output p and the marginal cost of the p th output. PSE p AICp C(Y) Y p (2.4) 14

31 Estimates of economies of scope (SC) represent the cost savings of producing multiple outputs within a single firm versus producing outputs individually. Economies of scope may be expressed in the following manner where C(Y) is total production cost, C(Y T ) is the cost of producing output Y T, and C(Y N-T ) represents the cost of producing the remaining outputs where Y N-T = (Y 1, Y k-1,0 0). CY ( T) CY ( NT) CY ( ) SC( Y) (2.5) CY ( ) Measures of multi-product economies of scale, product-specific economies of scale and economies of scope are related. The relationship between multi-product scale economies (MPSE), product-specific scale economies (PSE), and economies of scope (SC) can be determined by defining: i CY ( ) Yi Y i CY ( ) Yi Y i N i1 (2.6) where α i is the weight placed on the PSE of interest based upon its relative contribution to total output. Thus: ipsei( Y ) (1 i) PSENi( Y ) MPSE 1 SC( Y ) (2.7) MPSE can take one of three values: decreasing, constant or increasing returns to scale. Equation 2.7 examines the relationship among factors affecting MPSE. If SC(Y) is zero and the numerator is less than 1, equal to 1 or greater than 1, then there are decreasing, constant and 15

32 increasing returns to scale. If SC(Y) is greater than zero and the PSEs are at constant returns to scale, MPSE is in a region of increasing returns (>1). Data and Methods The Nonparametric Method To estimate the new scale measures, the cost (C i ) is determined for each firm following Färe, Grosskopf, and Lovell where costs are minimized for a given vector of input prices (w i ) and outputs (y i ) with the choice being the optimal input bundle (x * i ). min Ci w x st. Xz x ' yz 1 2 * i i ' * i i z z... z 1 z i y n (2.8) where there are n producers. The vector Z represents the weight of a particular firm with the sum of Z i s equal to 1 for variable returns to scale. From the above model, the costs and output quantities can be estimated. The output quantities (y i ) constrain the cost minimizing input bundle to be at or below that observed in the data. Total cost from the model (C i ) is the solution to the cost minimization problem including the production of all outputs for the i th firm. The cost of producing all outputs except one (C i,all-p ) where p is the dropped output and is determined by either forcing one of the outputs to equal zero or by dropping one of the p th output constraints. 16

33 To calculate multi-product scale measures, marginal costs must be determined. The marginal costs (MC i,p ) for the p th output are obtained from the shadow prices on the output constraints on the base model (equation 2.8). The calculation of multi-product economies of scale (MPSE) uses the total cost of producing all outputs (C i,all ), the marginal costs (MC i,p ), and the output levels produced (Y i,p ) (equation 2.1). There is an issue with the nonparametric marginal cost because the linear structure results in Kink Points on the frontier that results in non-unique marginal costs. Thus, the marginal costs for the most efficient firms may not be unique. In practice this is usually a relatively small number of firms. In addition, a range of estimates of marginal costs can be calculated. Product specific economies of scale (PSE) require the calculation of incremental costs (IC i,p ) which are the cost of producing all outputs minus the sum of the costs of all individual outputs except output (p) for firm i (equation 2.2). Previous methods to calculate incremental costs using the nonparametric method drop one or more of the output constraints from equation 2.8 to determine the cost of producing the output alone (Chavas and Aliber). For example if a firm produces four different products, four different linear programs would be estimated excluding one of the outputs at a time. In this research, results from dropping one of the output constraints are compared with constraining the appropriate output to zero. Using equation 2.2, average incremental costs (AIC i,p ) are determined by dividing incremental costs by individual output as shown in equation 2.3. From the average incremental cost (equation 2.3) and the marginal cost calculations from the shadow prices, it is possible to calculate PSEs (equation 2.4) where PSEs are interpreted similar to MPSEs except that PSEs pertain to only one output. 17

34 The calculation for scope economies (SC i ) follows from equation 2.5 where C i,p is the cost of producing output p for firm i, and C i,all is the cost of joint production of all outputs for firm i. This measure identifies the potential for cost savings through product diversification. Generally, SC i > 0 implies that scope economies exist and average per-unit costs are reduced with diversification. A scope measure of 0.5 implies that jointly producing multiple outputs in a two goods case would reduce costs of producing these outputs by 50% compared to producing them individually. Cost efficiency (CE) identifies a firm s proximity to the cost frontier for a given input/output bundle. It is the quotient of the estimated frontier cost (C i ) and the actual total cost (ATC) the firm incurred producing their output bundle. CE i C i ATC i (2.9) This measure must be greater than 0 but less than or equal to 1. A cost efficiency of 1 implies that the firm is operating on the frontier at the lowest possible cost for a given output bundle. However, a cost efficiency less than 1 implies that cost can be reduced by altering the input bundle. This sub-section has operationalized the measure of marginal costs and incremental costs necessary for the measurement of multi-product and product-specific scale economies. The next section examines the methods used to compare the accuracy of the nonparametric measures with those from a true cost frontier. Data Simulation The data for the analysis were generated utilizing a modified Monte Carlo procedure found in Gao and Featherstone (2008) run on the SHAZAM software platform with the code 18

35 found in Appendix A. A normalized quadratic cost function involving 3 inputs (x 1, x 2, x 3 ) with corresponding prices (w 1, w 2, w 3 ), and 2 outputs (y 1, y 2 ) with corresponding prices (p 1, p 2 ) was used. The normalized quadratic cost/ profit function is used because it is a self-dual cost function and a flexible functional form (Lusk et al.). The input and output prices (w i, p i ) are randomly generated following a normal distribution. Assumed distributions for the output prices and input prices provide observed prices strictly greater than zero with different means and standard deviations to ensure some variability in input/output quantity demands and relative prices. They are: w 1 ~ N (9, 0.99) w 2 ~ N (18, 1.98) w 3 ~ N (7, 0.77) (2.10) p 1 ~ N (325, 99) p 2 ~ N (800, 99) The input price variability was set proportionate to its mean while the output prices have different relative variability to represent products in markets with different volatilities. The outputs (y i ) and inputs (x j ) are determined as a function of input and output prices using an assumed underlying production technology. All prices are normalized on w 3 and the cost function is divided by w 3 to impose homogeneity. To ensure the curvature condition is met, the true cost function is assumed to be concave in input prices and convex in output quantities. The assumed parameters are set to satisfy the following theory based condition: b ij =b ji (symmetry in input prices). The assumed parameters (Table 2.1) are used to generate the output quantities y 1 and y 2 1. The general form of the normalized quadratic cost function is: 1 The analysis also was completed for alternative assumptions on price distributions. 19

36 w1 y1 CWY (, ) b0 b1 b2 a1 a2 w 2 y 2 1 b11 b12 w1 c11 c12 y1 a11 a12 y1 w1 w2 y1 y2 w1 w2 2 b21 b 22 w 2 c21 c 22 y 2 a21 a 22 y 2 (2.11) Output quantities (shown below) are calculated using the assumed parameters of the cost function (Table 2.1) and the random prices defined in equation ) ( c22c11 c12c12 ) ( c c c c c p c p a c a c w a c a c w a c ac y y c p c p a c a c w a c a c w a c ac (2.12) Using the above cost function (Equation 2.11), a positive random cost deviation term is added to the cost function following a half-normal distribution that alters the cost efficiency where the absolute value of e is distributed e~n (0,1000) 2. The inclusion of this term adds cost inefficiencies in the data such that firms are off the frontier effectively increasing their cost while keeping the output quantities the same. The level of inefficiency is half-normally distributed. An additional data set 3 is generated assuming a uniform distribution. The uniform deviation ranged from zero to 900. The normal distribution standard deviation of 1,000 generates a mean and standard deviation for cost efficiency roughly equivalent to a uniform distribution with a range from zero to 900. From equation (2.11), and using Shephard s Lemma where ( C(W,Y)/ w i )=x i, the factor demands for inputs x 1 and x 2 are recovered. Factor demand for x 3 is found by subtracting the product of quantities and prices for x 2 and x 3 from the total cost (equation 2.13). 2 The analysis also examined alternative normal standard deviations. 3 The analysis was run using 2500 observations with little difference in the results. 20

37 x b b w b w a y a y x b b w b w a y a y x C W, Y e x w x w ) (2.13) The input quantities (x i s) are then adjusted (x i a ) by the cost efficiency (CE) effectively increasing the input demands proportionate to the costs generated for each firm. x a i x CE i (2.14) Using the above method, 400 observations were generated where firms produce a combination of both outputs. Fifty firms were generated producing only y 1 with another 50 firms producing only y 2 which is accomplished by restricting either y 1 or y 2 to equal zero and rerunning the simulation for 50 separate observations each. Thus, a total of 500 observations were n simulated with descriptive statistics shown in Table 2.2. In Table 2.2, x i represents inefficient input quantities for the normal error distribution and x u i represent the inefficient input quantities for the uniform distribution. The summary statistics for the multi-product scale, product-specific scale, scope, and cost efficiencies for each data point from the true cost function are shown in Table 2.3. Summary statistics on scale and scope are independent of the distribution of cost inefficiency. Figures 2.1 through 2.4 provide visual representations of the multi-product scale and scope economies as well as cost efficiencies and product-specific scale economies calculated from the true cost function. These calculations are used to examine the accuracy of the proposed nonparametric approach. While the cost efficiency for each firm is simulated under a uniform and a half-normal distribution (Figure 2.2), the MPSE, PSE s, and Economies of Scope are identical for each data point (Table 2.3) for the true cost function. This is due to the fact that the input prices (w i s) and output prices (p i s) remain unchanged and thus, the output quantities (y i s) remain unchanged 21

38 (Equation 2.14). The input quantities (x i s) are different in that the deviation in input quantity is uniformly distributed. In the uniformly distributed data more evenly distributes the quantity of firms at each relative distance from the frontier, rather than many firms being clustered around the mean distance as in the half-normal case. The difference between the true and the nonparametric approach is evaluated by subtracting each nonparametric calculation from the true measure calculated with Monte Carlo simulation. Since the approximation of the true measure is key, the statistics reported hereafter are the difference between the true measures and what was estimated nonparametrically. Using this approach, any possible bias from the nonparametric approach can be determined. A positive number implies that the nonparametric approach underestimates the measure being evaluated and conversely, a negative difference indicates the nonparametric method overestimates the measure. The mean absolute deviation is also reported for all three models allowing for the comparison of average absolute deviation from zero Cumulative density functions are presented for the differences between the true measures and the estimated measures to produce visual representation of both bias and deviation. If there is no difference between the estimated measure and the true measure, the cumulative density function is a vertical line at zero (see figure 2.10 for the No Inefficiencies model). Results Three comparisons were conducted using the half-normal distribution for cost inefficiency, and three identical comparisons using the uniform distribution for cost inefficiency. The first comparison for both distributions uses the Monte Carlo data with only cost inefficiencies in the cost function (No Inefficiency). The purpose of this simulation is to ensure the model is estimating the measures correctly, and to examine the nonparametric procedure 22

39 estimates of scale and scope when all firms are efficient in input quantities. The second and third comparisons for both distributions involve introducing technical inefficiencies into the input quantities (equation 2.16), and are more consistent with observed data. Since efficient firms have a cost efficiency of 1, and less cost efficient firms have a cost efficiency between 0 and 1, an efficient firm uses optimal input quantities. However firms may use additional inputs to produce output if the firm is not efficient. Inputs x 1, x 2, and x 3 are adjusted upwards by the proportionate cost inefficiency to reflect this. The second nonparametric comparison for the half-normal and uniform distributions assume the appropriate constraints are dropped (Dropped) for the estimation of incremental costs. The third simulation forces the appropriate output to be 0 (Constrained). The estimation was done using the General Algebraic Modeling Software and the code can be found in Appendix A. Twenty-four frontier points are identified from the nonparametric estimates for the halfnormal distribution and twenty-five using the uniform data. For each distribution, the firms found on the frontier were the same for the Dropped Model and the Constrained Model. These points have non-unique marginal cost estimates. Due to the non-uniqueness of the marginal costs from these observations, MPSEs cannot be calculated. For single output observations, PSEs cannot be calculated for the output not being produced. Economies of scope are also not calculated for single output observations. Multi-product Economies of Scale The differences results for MPSE are found in Table 2.4 and Figure 2.5. The No Inefficiencies model for both distributions shows little difference from the actual frontier function (Figure 2.5). The average bias was close to 0 for both distributions with a standard 23